Degenerations of generalized Kummer varieties
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917413622120448 |
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| author | Halle, Lars H. Hulek, Klaus Zhang, Ziyu |
| author_facet | Halle, Lars H. Hulek, Klaus Zhang, Ziyu |
| contents | We present a method to construct explicit degenerations of higher-dimensional generalized Kummer varieties. We start with a simple degeneration $f: \mathcal Y \to C$ of abelian surfaces. Then $ \mathcal{Y} \setminus \mathcal{Y}_0$ is an abelian scheme over $C \setminus 0$ and we can form the relative generalized Kummer variety $K^{n-1}_{\circ} = \mathrm{Kum}^{n-1}(\mathcal{Y} \setminus \mathcal{Y}_0) \to C \setminus 0$. This is naturally a closed subscheme of the relative Hilbert scheme $\mathrm{Hilb}^{n}(\mathcal{Y} \setminus \mathcal{Y}_0) \to C \setminus 0$. In previous work (joint with Gulbrandsen) we had constructed a compactification $I^n_{\mathcal{Y}/C}$ over $C$ of the latter scheme. The closure $K^{n-1}_{\mathcal{Y}/C}$ of $K^{n-1}_{\circ}$ inside $I^n_{\mathcal{Y}/C}$ yields a canonical way to degenerate the family of generalized Kummer varieties, and is the degeneration we propose. This paper contains a detailed study of the geometry of the scheme $K^{n-1}_{\mathcal{Y}/C}$ and its natural stratification. For $n=2$ we obtain a projective Kulikov model of Kummer surfaces, whereas already for $n=3$ new phenomena occur. We study in detail the dual complex of $K^{2}_{\mathcal{Y}/C}$ and show that this is PL-homeomorphic to the standard $2$-simplex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14890 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Degenerations of generalized Kummer varieties Halle, Lars H. Hulek, Klaus Zhang, Ziyu Algebraic Geometry Primary: 14D06, Secondary: 14C05, 14D23 We present a method to construct explicit degenerations of higher-dimensional generalized Kummer varieties. We start with a simple degeneration $f: \mathcal Y \to C$ of abelian surfaces. Then $ \mathcal{Y} \setminus \mathcal{Y}_0$ is an abelian scheme over $C \setminus 0$ and we can form the relative generalized Kummer variety $K^{n-1}_{\circ} = \mathrm{Kum}^{n-1}(\mathcal{Y} \setminus \mathcal{Y}_0) \to C \setminus 0$. This is naturally a closed subscheme of the relative Hilbert scheme $\mathrm{Hilb}^{n}(\mathcal{Y} \setminus \mathcal{Y}_0) \to C \setminus 0$. In previous work (joint with Gulbrandsen) we had constructed a compactification $I^n_{\mathcal{Y}/C}$ over $C$ of the latter scheme. The closure $K^{n-1}_{\mathcal{Y}/C}$ of $K^{n-1}_{\circ}$ inside $I^n_{\mathcal{Y}/C}$ yields a canonical way to degenerate the family of generalized Kummer varieties, and is the degeneration we propose. This paper contains a detailed study of the geometry of the scheme $K^{n-1}_{\mathcal{Y}/C}$ and its natural stratification. For $n=2$ we obtain a projective Kulikov model of Kummer surfaces, whereas already for $n=3$ new phenomena occur. We study in detail the dual complex of $K^{2}_{\mathcal{Y}/C}$ and show that this is PL-homeomorphic to the standard $2$-simplex. |
| title | Degenerations of generalized Kummer varieties |
| topic | Algebraic Geometry Primary: 14D06, Secondary: 14C05, 14D23 |
| url | https://arxiv.org/abs/2604.14890 |